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On the continuity of the tangent cone to the determinantal variety

Guillaume Olikier, P. -A. Absil

TL;DR

This work analyzes the continuity of the tangent cone map to the rank-constrained determinantal set $\mathbb{R}_{\le r}^{m\times n}$ by explicitly characterizing inner and outer limits of the tangent-cone correspondence as matrices converge to a given rank. It establishes a chain of inclusions for these limits and identifies conditions under which the map is continuous versus merely semicontinuous, with a parallel analysis for the associated normal cones. The results provide precise, rank-dependent continuity statements and connect these geometric insights to $a$-regularity of Whitney stratifications of real determinantal varieties. The findings have direct implications for the convergence analysis of rank-adaptive optimization algorithms operating on low-rank matrix sets and offer a rigorous bridge between variational geometry and real-algebraic stratifications.

Abstract

Tangent and normal cones play an important role in constrained optimization to describe admissible search directions and, in particular, to formulate optimality conditions. They notably appear in various recent algorithms for both smooth and nonsmooth low-rank optimization where the feasible set is the set $\mathbb{R}_{\leq r}^{m \times n}$ of all $m \times n$ real matrices of rank at most $r$. In this paper, motivated by the convergence analysis of such algorithms, we study, by computing inner and outer limits, the continuity of the correspondence that maps each $X \in \mathbb{R}_{\leq r}^{m \times n}$ to the tangent cone to $\mathbb{R}_{\leq r}^{m \times n}$ at $X$. We also deduce results about the continuity of the corresponding normal cone correspondence. Finally, we show that our results include as a particular case the $a$-regularity of the Whitney stratification of $\mathbb{R}_{\leq r}^{m \times n}$ following from the fact that this set is a real algebraic variety, called the real determinantal variety.

On the continuity of the tangent cone to the determinantal variety

TL;DR

This work analyzes the continuity of the tangent cone map to the rank-constrained determinantal set by explicitly characterizing inner and outer limits of the tangent-cone correspondence as matrices converge to a given rank. It establishes a chain of inclusions for these limits and identifies conditions under which the map is continuous versus merely semicontinuous, with a parallel analysis for the associated normal cones. The results provide precise, rank-dependent continuity statements and connect these geometric insights to -regularity of Whitney stratifications of real determinantal varieties. The findings have direct implications for the convergence analysis of rank-adaptive optimization algorithms operating on low-rank matrix sets and offer a rigorous bridge between variational geometry and real-algebraic stratifications.

Abstract

Tangent and normal cones play an important role in constrained optimization to describe admissible search directions and, in particular, to formulate optimality conditions. They notably appear in various recent algorithms for both smooth and nonsmooth low-rank optimization where the feasible set is the set of all real matrices of rank at most . In this paper, motivated by the convergence analysis of such algorithms, we study, by computing inner and outer limits, the continuity of the correspondence that maps each to the tangent cone to at . We also deduce results about the continuity of the corresponding normal cone correspondence. Finally, we show that our results include as a particular case the -regularity of the Whitney stratification of following from the fact that this set is a real algebraic variety, called the real determinantal variety.
Paper Structure (15 sections, 21 theorems, 61 equations, 1 figure, 1 table)

This paper contains 15 sections, 21 theorems, 61 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Euclidean vector space of real matrices and some of its submanifolds
  4. Inner and outer limits, continuity of correspondences
  5. Tangent and normal cones
  6. Tangent and normal cones to the real determinantal variety
  7. On the rank of $2 \times 2$ block matrices
  8. Main result
  9. Proof of Theorem \ref{['theorem:MainResult']}
  10. Complementary results
  11. Inner and outer limits of ${T}_{\mathbb{R}_{\le \mkern 0.9mu\overline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu}^{m \times n}}^{\mathrm{C}}(\cdot)$ relative to $\mathbb{R}_r^{m \times n}$ and $\mathbb{R}_{\le r}^{m \times n}$
  12. Continuity of the normal cones to $\mathbb{R}_{\le \mkern 0.9mu\overline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu}^{m \times n}$ relative to $\mathbb{R}_{\le r}^{m \times n}$ on $\mathbb{R}_r^{m \times n}$
  13. Inner limits of the normal cones to $\mathbb{R}_{\le \mkern 0.9mu\overline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu}^{m \times n}$ relative to $\mathbb{R}_r^{m \times n}$ at points of $\mathbb{R}_{< r}^{m \times n}$
  14. Outer limits of the normal cones to $\mathbb{R}_{\le \mkern 0.9mu\overline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu}^{m \times n}$ relative to $\mathbb{R}_r^{m \times n}$ at points of $\mathbb{R}_{< r}^{m \times n}$
  15. Connection with the $a$-regularity of the Whitney stratification

Key Result

Proposition 2.1

For every positive integer $r \le \min\{m,n\}$, $\mathbb{R}_r^{m \times n}$ has the following properties:

Figures (1)

  • Figure 1: Five tangent or normal cones to a locally closed subset $\mathcal{S}$ of $\mathbb{R}^{m \times n}$.

Theorems & Definitions (41)

  • Proposition 2.1
  • proof
  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • Lemma 4.1
  • ...and 31 more